Method and apparatus for estimating error in multi-axis controlled machine

ABSTRACT

A method and apparatus for estimating error in a multi-axis controlled machine is applicable to any type of machine configuration in order to estimate and confirm in advance the final position and the posture of the machine, which are produced when geometric errors of the machine are synthesized. The method includes defining the structure of the multi-axis controlled machine subjected to error estimation; and defining parameters, which represent behaviors of drive axes having geometric error and relationships between the drive axes according to the defined structure of the multi-axis controlled machine, adding the defined parameters by applying the parameters to a generalized error synthesis model, and generating an error synthesis model of the multi-axis controlled machine by applying a result of parametric modeling in response to a result of the adding.

CROSS REFERENCE TO RELATED APPLICATION

The present application claims priority from Korean Patent ApplicationNumber 10-2010-0018364 filed on Mar. 2, 2010, the entire contents ofwhich application is incorporated herein for all purposes by thisreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method and apparatus for estimatingerror in a multi-axis controlled machine, and more particularly,applicable to any type of machine configurations in order to estimateand verify in advance the final position and posture of the machineaffected by perplexedly interlinked geometric errors.

2. Description of Related Art

Multi-axis controlled machines are a type of mechanical apparatus thatincludes two or more drive axes. A multi-axis controlled machine can bea multi-axis machine tool that has a combination of a number of linearor rotary drive axes, such as a linear slide or a rotary or tiltingtable, a multi-axis articulated robot, a Coordinate Measuring Machine(CMM), or the like. FIG. 1 shows a five-axis machine tool as an exampleof a multi-axis controlled machine.

The multi-axis controlled machines have many geometrical errors, such aserror of each drive axis and error between drive axes, which areperplexedly interlinked to each other, thereby resulting in thedeviation of final posture (i.e., position and orientation) when themulti-axis controlled machine is driven. Various error synthesis models,as mathematical expressions, are derived depending on the structures andshapes of the multi-axis machines.

Meanwhile, in the stage of designing or fabricating multi-axiscontrolled machines, it is necessary to measure and evaluate geometricerrors of the machines in order to verify the performance and compensatefor the errors of the machines. Devices used to measure such errorsgenerally include a laser interferometer, a ball bar, a capacitancesensor, an autocollimator, a Position-Sensitive Detector (PSD), and thelike. In particular, a variety of measuring methods has been studied anddeveloped for multi-axis controlled machines having rotary axes due tothe difficulty involved in measurement.

In addition, in order to estimate the performance of the multi-axiscontrolled machines, it is necessary to evaluate the final error effectsof a number of geometric errors. However, in the multi-axis controlledmachines having different configurations, technologies that define therelationships between a number of geometric errors, such as the error ofa drive axis and the error between drive axes, and derive final erroreffects have not been automated yet. Thus, complicated and difficultmathematical calculation has to be performed every time, according tothe configuration of a multi-axis controlled machine. In addition, itbecomes very difficult for engineers to directly evaluate thecontribution of respective errors to the final posture error. Inparticular, the greater the number of drive axes, the more significantthese problems are.

Therefore, conventional error estimation technologies, which are used ingeneral industrial sites, can only measure and estimate a small numberof errors of a single axis or a small number of axes using an errorestimation device, or estimate the performance of processing the errorsby measuring the final result obtained by directly processing theerrors. In addition, the accuracy of estimation is low.

The information disclosed in this Background of the Invention section isonly for the enhancement of understanding of the background of theinvention and should not be taken as an acknowledgment or any form ofsuggestion that this information forms a prior art that would already beknown to a person skilled in the art.

BRIEF SUMMARY OF THE INVENTION

Various aspects of the present invention provide a method and apparatusfor estimating errors in a multi-axis controlled machine that isapplicable to any type of machine configurations and can estimate andconfirm posture error in a final result in advance, which is generatedwhen geometric errors of the multi-axis controlled machine are combined,without direct processing or a change in position.

In an aspect of the present invention, the method for estimating errorin a multi-axis controlled machine may include the processes of:defining the configuration of the multi-axis controlled machinesubjected to error estimation; and defining parameters, which representbehaviors of drive axes having geometric error and relationships betweenthe drive axes according to the defined structure of the multi-axiscontrolled machine, adding the defined parameters by applying theparameters to a generalized error synthesis model, and generating anerror synthesis model of the multi-axis controlled machine by applying aresult of parametric modeling in response to a result of the adding.

According to an exemplary embodiment of the invention, the process ofdefining the structure of the multi-axis controlled machine can defineone or more selected axis from the group consisting of defining driveaxes of the multi-axis controlled machine, a posture of a tool, offsetdistances between the drive axes, and a setting of squareness error.

According to an exemplary embodiment of the invention, the process ofdefining the drive axes of the multi-axis controlled machine can includethe step of sequentially arranging the drive axes, which are oriented inforward and backward directions in a reference coordinate system of themulti-axis controlled machine.

According to an exemplary embodiment of the invention, the process ofgenerating an error synthesis model can include the step of determininglocal coordinate systems each link of the multi-axis controlled machine;defining parameters, which represent the behaviors of the drive axeshaving geometric error and the relationships between the drive axesaccording to the defined structure of the multi-axis controlled machine;adding the defined parameters by applying the parameters to thegeneralized error synthesis model; performing parametric modeling onerror measurement data, which are measured from the multi-axiscontrolled machine; and generating the error synthesis model of themulti-axis controlled machine by applying the result of the parametricmodeling in response to the result of the adding.

According to an exemplary embodiment of the invention, the generalizederror synthesis model is derived by selecting one or more from the groupconsisting of kinematic error modeling of linear and rotary axes;modeling one or more drive axes by arranging the drive axes in forwardor backward direction with respect to a reference coordinate system; andexclusion higher order terms of errors in the kinematic error modeling,which includes arranging the drive axes.

According to an exemplary embodiment of the invention, the generalizederror synthesis model can be expressed by formulas:

According to an exemplary embodiment of the invention, the method mayfurther include processes of: creating a virtual coordinate of a point,which includes error of a designated shape to be processed or at aposition to be confirmed, using the error synthesis model; andgenerating an error map, which visually expresses the created imaginarypoint.

According to an exemplary embodiment of the invention, the process ofcreating a virtual coordinate can include the steps of: determining thedesignated shape to be processed and the position to be confirmed;generating a tool path for the designated shape to be processed or theposition to be confirmed; generating or inputting a motion parameterabout each of the drive axes of the multi-axis controlled machine alongthe tool path of the multi-axis controlled machine; and creating avirtual coordinate having error for each of points in the tool pathusing the error synthesis model.

According to an exemplary embodiment of the invention, the designatedshape to be processed can be a frustum of a cone or a hemisphere.

In an aspect of the present invention, the apparatus for estimatingerror in a multi-axis controlled machine may include a mechanisticstructure definition module for defining a mechanistic structuresubjected to error estimation; an error synthesis module for definingparameters, which represent behaviors of drive axes having geometricerror and relationships between the drive axes according to the definedstructure of the multi-axis controlled machine, adding the definedparameters by applying the parameters to a generalized error synthesismodel, and generating an error synthesis model of the multi-axiscontrolled machine by applying a result of parametric modeling inresponse to a result of the adding; and an error mapping module forcreating a virtual coordinate, which includes error of a designatedshape to be processed or at a position to be confirmed, using the errorsynthesis model, and generating an error map, which visually expressesthe created imaginary point.

According to an exemplary embodiment of the invention, the machineconfiguration definition module can include one or more selected fromthe group consisting of defining the drive axes of the multi-axiscontrolled machine, defining a posture of a tool, defining setting thesquareness, and offset distances between the drive axes error.

According to an exemplary embodiment of the invention, the mechanisticstructure definition module can define the drive axes of the multi-axiscontrolled machine by sequentially arranging the drive axes, which areoriented in forward and backward directions in a reference coordinatesystem of the multi-axis controlled machine.

According to an exemplary embodiment of the invention, the generalizederror synthesis model is expressed by formulas:

According to an exemplary embodiment of the invention, the designatedshape to be processed can be a frustum of a cone or a hemisphere.

According to exemplary embodiments of the present invention as set forthabove, the method and apparatus for estimating error in a multi-axiscontrolled machine can evaluate geometric errors, which occur in a finalresult when the multi-axis controlled machine actually operates ormoves, by calculating error vectors including geometric errors withrespect to any shape of the orientations of points on a tool path orpositioning vectors, such as a frustum of a cone or a hemisphere,without directly operating or moving the multi-axis controlled machine.In addition, since the method and apparatus is applicable to any typesof multi-axis controlled machine having different configurations, themethod and apparatus can be used for estimating the performance of acarrier system and in a simulation operation of an error compensationsystem.

The methods and apparatuses of the present invention have other featuresand advantages which will be apparent from, or are set forth in greaterdetail in the accompanying drawings, which are incorporated herein, andin the following Detailed Description of the Invention, which togetherserve to explain certain principles of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view showing an example of a multi-axiscontrolled machine;

FIG. 2 is a diagram showing local coordinate systems according to anexemplary embodiment of the invention, which are set to respective driveaxes of the multi-axis controlled machine shown in FIG. 1;

FIG. 3 is a diagram explaining an offset from among various parameters,which express the relationships between the local coordinate systems,according to an exemplary embodiment of the invention;

FIG. 4 is a diagram showing least squares lines, which are calculated onthe basis of actual routes measured on three linear axes, according toan exemplary embodiment of the invention;

FIG. 5 is a diagram explaining squareness error about the routes alongthe linear axes, shown in FIG. 4, according to an exemplary embodimentof the invention;

FIG. 6 is a diagram showing a least squares center and a central axis,calculated with respect to the translation of a table when a rotary axisis operating, according to an exemplary embodiment of the invention;

FIG. 7 is a diagram explaining the squareness and the offset error ofthe rotary axis, shown in FIG. 6, according to an exemplary embodimentof the invention;

FIG. 8 is a diagram explaining position error and angular error fromamong various parameters, which express the relationships between localcoordinate systems, according to an exemplary embodiment of theinvention;

FIGS. 9A and 9B are diagrams explaining kinematic modeling of the linearand rotary axes in the multi-axis controlled machine;

FIG. 10 a is a diagram showing an illustrative structure in which anumber of drive axes are connected in forward direction with respect toa reference coordinate system in the multi-axis controlled machine;

FIG. 10 b is a diagram showing an illustrative structure in which anumber of drive axes are connected in inverse direction with respect toa reference coordinate system in the multi-axis controlled machine;

FIG. 11 is a block diagram showing the apparatus for estimating error ina multi-axis controlled machine according to an exemplary embodiment ofthe invention;

FIG. 12 is a flowchart showing the entire process of the method forestimating error in a multi-axis controlled machine according to anexemplary embodiment of the invention;

FIG. 13 is a flow diagram showing error synthesis modeling in detailaccording to an exemplary embodiment of the invention;

FIG. 14 is a flow diagram showing error mapping in detail according toan exemplary embodiment of the invention;

FIG. 15A is a table showing an example of a process of inputting errormeasurement data according to an exemplary embodiment of the invention;

FIG. 15B is a table showing an example of motion parameter input valuesof the error measurement data according to an exemplary embodiment ofthe invention;

FIG. 16 is a diagram showing an example of an error map generated withrespect to one machine coordinate according to an exemplary embodimentof the invention; and

FIGS. 17 to 19 are diagrams showing examples of error maps generatedwith respect to a plurality of machine coordinates according to anexemplary embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Reference will now be made in detail to various embodiments of thepresent invention(s), examples of which are illustrated in theaccompanying drawings and described below. While the invention(s) willbe described in conjunction with exemplary embodiments, it will beunderstood that the present description is not intended to limit theinvention(s) to those exemplary embodiments. On the contrary, theinvention(s) is/are intended to cover not only the exemplaryembodiments, but also various alternatives, modifications, equivalentsand other embodiments that may be included within the spirit and scopeof the invention as defined by the appended claims.

In the following description of the present invention, detaileddescriptions of known functions and components incorporated herein willbe omitted when they may make the subject matter of the presentinvention unclear.

In addition, throughout this document, reference should be made to thedrawings, in which the same reference numerals and signs are usedthroughout the different drawings to designate the same or similarcomponents

In addition, throughout this specification and the claims that follow,it will be understood that when an element is referred to as being“connected to” or “coupled to” another element, not only can it be“directly connected or coupled to” the other element, but also can it be“indirectly connected or coupled to” the other element via anintervening element. Unless explicitly stated to the contrary, the word“comprise” and its variations, such as “comprises” or “comprising” willbe understood to imply the inclusion of stated elements but not theexclusion of any other elements.

Furthermore, the terminology “module” should be understood as indicatinga unit that conducts a specific function or operation. The module can beembodied as a piece of hardware or software or a combination of hardwareand software.

Basic Principle

FIG. 1 is a perspective view showing an example of a multi-axiscontrolled machine to which the present invention is applicable.Specifically, FIG. 1 shows a multi-axis machine tool that serves toprocess a workpiece having any shape.

The machine tool shown in FIG. 1 is an RRTFTT-type five-axis machinetool that has two rotary axes and three linear axes. Two drive axes,that is, X and Z axes, are used to displace the tool, whereas Y, A, andC axes are used to displace and rotate the workpiece. In other words,the tool linearly moves along the X and Z axes, whereas the workpiecetilts about the X axis using the A axis and rotates about the Z axisusing the C axis while linearly moving along the Y axis. In thefollowing description, the construction and operation of a method andapparatus for estimating error in a multi-axis controlled machineaccording to an exemplary embodiment of the invention will be describedwith reference to the five-axis machine tool by way of example.

Below, a description will first be given on basic principles andterminologies with respect to the method and apparatus for estimatingerror in a multi-axis controlled machine according to an exemplaryembodiment of the invention.

Homogeneous Transform Matrix (HTM)

According to an exemplary embodiment of the present invention, an HTM isused in order to effectively express links and states of errors in anumber of drive axes of a multi-axis controlled machine. The HTM canexpress the posture between two coordinate systems, using rotation (ROT)and translation (TRS) with respect to the posture. For this, the HTMaccording to an exemplary of the invention can be expressed as follows:

The HTM is a matrix that expresses the posture between {i} coordinatesystem and {i+1} coordinate system, in which ROT is expressed as a 3×3matrix, indicating the orientation of the {i+1} coordinate system viewedfrom the {i} coordinate system, and TRS is expressed as a 3×1 vector,indicating the position of that point.

Definition of Local Coordinate System

In addition, according to an exemplary embodiment of the invention, alocal coordinate system on a respective drive axis (i.e., a linear orrotary axis) is defined on the basis of the home position of themulti-axis controlled machine. The local coordinate system of theRRFTTT-type five-axis machine tool, shown in FIG. 1, can be defined asshown in FIG. 2. Since the local coordinate system of each axis is setin the same orientation as a reference coordinate system is, the sign ofa drive axis indicated by a machine controller can be different fromthat of the local coordinate system. For example, solid arrows in FIG. 2indicates actual directions of translation, in which the actualdirections of translation of X and Z axes are defined to be opposite tothe signs of axes of the respective local coordinate systems.

In addition, the positions of the respective local coordinate systemsare defined according to the structure of the machine. In the case ofthe machine shown in FIGS. 1 and 2, when all of the drive axes arelocated at their home positions, local coordinate systems {X}, {Y}, and{Z} of linear drive axes are located at the terminal end of a spindle orthe terminal end of a tool like a tool coordinate system {T}, and localcoordinate systems {A} and {C} of rotary drive axes are located at thepoint where the centers of rotation of the two axes meet. Therefore, asshown in FIG. 2, the centers of the linear and rotary axes are offsetfrom each other.

The position of the reference coordinate system {R} is set so that it isconsistent with that of the local coordinate systems {X}, {Y}, and {Z}of linear drive axes, and the orientation of the reference coordinatesystem is set in consideration of straight lines, which are producedfrom geometric errors and actual routes of translation of three linearaxes using the least squares method. Then, the squareness error betweenthe individual linear axes is calculated on the basis of thisorientation.

Definition of Parameters

In addition, according to an exemplary embodiment of the invention,parameters are defined, as reported in Table 1 below, in order toexpress the relationships between the local coordinate systems ofrespective links of the multi-axis controlled machine. In Table 1 below,each parameter is expressed by a 4×4 HTM.

TABLE 1 Parameter symbol Title Remarks OM Offset Home position ininitial state with respect to previous coordinate system (includingerrors) SM Squareness Orientation error in initial state with respect toprevious coordination system (including initial orientation) TMTranslation Machine instruction of linear axes (X, Y, and Z axes) AMRotation Machine instruction of rotary table (A and C axes) DM Positionerror Position error at point of machine instruction EM Angular errorAngular error at point of machine instruction

In Table 1 above, OM and SM indicate the relationships in posture (i.e.,orientation and position) between two continuous local coordinatesystems, and TM and AM indicate machine input values for instructingoperation of the linear and rotary axes. In addition, DM and EM areposition-dependent geometric errors, which are different depending ontheir positions. DM and EM indicate error terms that are present inrespective positions of translation and rotation whenever an individualdrive axis is translated or rotated.

Below, a more detailed description will be given of the parameters.

The offset OM is a parameter that indicates the position informationbetween initially-defined local coordinate systems. In general, theoffset is not considered for a three-axis machine tool since the machinetool can be translated so that all of three axes can meet at one point.However, it has to be considered always for the multi-axis machine toolsince a rotary table includes an offset or offset error in the referencecoordinate system. Due to error that occurs during the actualfabrication and assembly of a machine, error in dimension and assemblyoccurs in addition to the design offset. Such error is referred to as“offset error.” When performing error synthesis modeling, the offset andthe offset error can be considered as a bundle since they mathematicallyrepresent the same position component. Specifically, it can beconsidered as: offset=offset+offset error. As shown in FIG. 3, theoffset between {i−1} coordinate system and the {i} coordinate system canbe expressed by a 4×4 HTM, as in Formula 3 below, and the positionvector of the HTM can be expressed separately using symbols.

The squareness SM is a parameter that expresses the orientation ofinitially-defined local coordinate systems. When the Formula 1,mentioned above, is used in the initial setting, it is preferred thatthe orientations of all of the local coordinate systems, if possible, beset to be consistent with the directions of translation of drive axes.If the orientation of the reference coordinate system is the same asthose of all of the local coordinate systems, the orientation componentsof the initial coordinate systems are expressed as a unit matrix.However, the squareness between the coordinate systems is inevitable dueto error that occurs during the fabrication or assembly. In addition,like the offset, this parameter can be expressed mathematically byadding the initial orientation information and the squareness component.Furthermore, the squareness can also be expressed as includingparallelism if a reference axis is considered differently.

That is, squareness=initial Orientation+squareness (parallelism). Ingeneral, because most of the local coordinate systems have a mechanicalstructure such that their initial orientations can be consistent withthat of the reference coordinate system, the apparatus according to anexemplary embodiment of the invention disregards the initial setting oforientation.

The method of defining the squareness due to the geometric errorseparately considers the linear axis and the rotary axis. The squarenessof the linear axis can be defined using least squares lines, which areobtained by applying the least squares method using points on the routesof actual translation. In the case of measuring routes of translation ofthe three linear axes on the basis of one specific coordinate system inFIG. 4, least squares lines can be produced by applying the leastsquares method to the routes of translation. The three least squareslines have a total of six errors in orientation, and as shown in FIG. 5,the squareness and the orientation of the reference coordinate systemcan be defined using the errors in orientation. For example, in amachine tool where {X}, {Y}, and {Z} local coordinate systemscorresponding to the three linear axes are set as {1}, {2}, and {3}, ifthe x axis of the reference coordinate system {R} is brought to beconsistent with the least squares line, corresponding to the route oftranslation of the {X} coordinate system, and the y axis of thereference coordinate system {R} is rotated about the x axis so as to beconsistent with the plane defined by the least squares lines of the {X}and {Y} coordinate systems, three orientation error components about thereference coordinate system {R} become the squareness.

Meanwhile, the squareness of the rotary axis is shown in FIGS. 6 and 7.As shown in FIG. 6, when the rotary table is operating, the leastsquares home position and the orientation of the central axis ofrotation can be produced by applying the least squares method to actualroutes of translation. If the local coordinate system is brought to beconsistent with the point where the imaginary center of rotation meetsthe plane of an ideal coordinate system {Ideal}, two orientation errorcomponents occur as squareness, as shown in FIG. 7, and two offseterrors are defined about the ideal coordinate system.

Therefore, the squareness SM between the coordinate systems can beexpressed mathematically using the rotation components of the coordinatesystem. Since the squareness component is a sufficiently-small value oferror, higher order terms can be omitted. Accordingly, the squareness SMcan be expressed as in Formula 2 below. In addition, if a sub-matrixcorresponding to the rotation component of a homogeneous transformationmatrix SM is decomposed by the sum of a 3×3 unit matrix, it can beexpressed by a skew matrix as in Formula 2 below.

In Formula 2 above, S_(ji) indicates a squareness error component whenthe {i} coordinate system is rotated about the j axis. For example,S_(zx) means squareness when the {X} coordinate system is rotated alongthe z axis, and indicates a squareness component defined by the xyplanes.

Translation TM and rotation AM are machine instructions, which are inputinto the multi-axis controlled machine. TM includes information on thedistance of linear translation, and AM includes information on the angleof rotation. TM and AM can be expressed by Formulae 3 and 4 below.

If a coordinate system is translated by an amount x along the X-axis,t_(x)=[x 0 0]^(T)If a coordinate system is translated by an amount y along the Y-axis,t_(y)=[0 y 0]^(T)If a coordinate system is translated by an amount z along the Z-axis,t_(z)=[0 0 z]^(T)

If a coordinate system is rotated by an amount a about the X-axis,

$A_{a} = \begin{bmatrix}1 & 0 & 0 \\0 & {\cos\; a} & {{- \sin}\; a} \\0 & {\sin\; a} & {\cos\; a}\end{bmatrix}$If a coordinate system is rotated by an amount b about the Y-axis,

$A_{b} = \begin{bmatrix}{\cos\; b} & 0 & {\sin\; b} \\0 & 1 & 0 \\{{- \sin}\; b} & 0 & {\cos\; b}\end{bmatrix}$If a coordinate system is rotated by an amount c about the Z-axis,

$A_{c} = \begin{bmatrix}{\cos\; c} & {{- \sin}\; c} & 0 \\{\sin\; c} & {\cos\; c} & 0 \\0 & 0 & 1\end{bmatrix}$

Here, the centers of rotation of a, b, and c axes are consistent withthe orientations of x, y, and z axes of the reference coordinate system.

Finally, when the linear axis or rotary axis operates in response to themachine instruction, each local coordinate system always accompanieswith geometric errors, which include position error DM and angular errorEM. These errors are also referred to as Position-Dependent GeometricError Parameters (PDGEP) since they are different depending on thepoints of respective machine instructions.

FIG. 8 illustrates an ideal final point, which occurs when the lineardrive axis operates according to the machine instruction, and an actualposition, which is changed by actual geometric error. In this case, theangular error EM and the position error DM can be expressed by Formulae5 and 6 below. (Refer to FIG. 6B as for the definition of the rotarydrive axis.) Since terms of angular errors are sufficiently-small valuesof errors like the squareness, higher order terms can be omitted

Here, ε_(ji) indicates rotational error components about the j axis,when the {i} coordinate system operate. The three rotational errorcomponents occurring from the three linear axes include “roll,” “pitch,”and “yaw.” Of the three errors occurring from the rotary drive axis, therotational error component along the central axis of rotation isreferred to as “angular positioning error” and the other two rotationalerror components, which are not with the central axis of rotation, arereferred to as “tilt errors.”

In addition, δ_(ji) indicates position error components in the j axialdirection when the {i} coordinate system operates. In the three linearaxes, if i=j, it is linear displacement error, and in the case of i≠j,it is straightness. Of the three errors of the rotary drive axis, axialerror occurs if the j direction coincides with the direction of thecentral axis of rotation, and radial error occurs if the j directiondoes not coincides with the direction of the central axis of rotation.

Method of Producing Generalized Error Synthesis Model

According to the invention, it is possible to produce an error synthesismodel using the parameters and coordinate systems, which are defined asabove. For this, a generalized error synthesis model, which can be usedwidely in the multi-axis controlled machine having any structure and anynumber of axes, is required. The generalized error synthesis model canbe designed as follows.

When individual drive axes of the multi-axis controlled machine aretranslated or rotated, a posture of each drive axis at one time pointcan be produced by kinematic error modeling. Here, the kinematic modelsof the linear and rotary axes are expressed by Formula 7 and 8 below.

$\begin{matrix}\begin{matrix}{{\tau_{i - 1}^{L} = {{OM}\mspace{14mu}{SM}\mspace{14mu}{TM}\mspace{11mu}{DM}\mspace{14mu}{EM}}}}_{i - 1}^{L} \\{= {{{{\left\lbrack {Io_{L}} \right\rbrack\left\lbrack {I + {S_{L}0}} \right\rbrack}\left\lbrack {It_{L}} \right\rbrack}\left\lbrack {Id_{L}} \right\rbrack}\left\lbrack {I + {E_{L}0}} \right\rbrack}}\end{matrix} & {{Formula}\mspace{14mu} 7} \\\begin{matrix}{{\tau_{i - 1}^{R} = {{OM}\mspace{14mu}{SM}\mspace{14mu}{DM}\mspace{11mu}{EM}\mspace{14mu}{AM}}}}_{i - 1}^{R} \\{= {{{{\left\lbrack {Io_{R}} \right\rbrack\left\lbrack {I + {S_{R}0}} \right\rbrack}\left\lbrack {Id_{R}} \right\rbrack}\left\lbrack {I + {E_{R}0}} \right\rbrack}\left\lbrack {A_{R}0} \right\rbrack}}\end{matrix} & {{Formula}\mspace{14mu} 8}\end{matrix}$

FIGS. 9A and 9B are diagrams explaining the kinematic modeling of thelinear and rotary axes. As in Formula 8, DM and EM in the model of therotary axis, unlike in the model of the linear axis, are defined on thebasis of a coordinate system that includes OM and SM regardless of therotary drive value.

It is required to separate the modeling of the linear axis (slide) fromthe modeling of the rotary axis (turntable/tilting table) in order torealize the generalized error synthesis model in the multi-axiscontrolled machine. For this purpose, a unified model for any drive axisis defined as follows, by combining the model of linear and rotary axis,which was described above.

Linear Axis:

$\left. \left. \begin{matrix}{{\tau_{i - 1}^{L} = {{OM}\mspace{14mu}{SM}\mspace{14mu}{TM}\mspace{11mu}{DM}\mspace{14mu}{EM}}}}_{i - 1}^{L} & \; \\{{Rotary}\mspace{14mu}{axis}\text{:}} & \; \\{{\tau_{i - 1}^{R} = {{OM}\mspace{14mu}{SM}\mspace{14mu}{DM}\mspace{11mu}{EM}\mspace{14mu}{AM}}}}_{i - 1}^{R} & \;\end{matrix} \right\rbrack\longrightarrow{\quad{\tau_{i - 1}^{A} = {{OM}\mspace{14mu}{SM}\mspace{14mu}{TM}\mspace{11mu}{DM}\mspace{14mu}{EM}\mspace{14mu}{AM}}}}_{i - 1}^{A} \right.\text{:}\mspace{14mu}{Arbitrary}\mspace{14mu}{axis}$

When a program is executed, the kinematic error model defined as aboveis divided into the linear axis part and the rotary axis part accordingto individual structures as below as soon as the shape and structure ofthe system are input. The linear axis part and the rotary axis part aredefined as below.

Meanwhile, the final kinematic error model of any axis, which is definedas above, is derived as in Formula 9 below.

Here, since errors have sufficiently-small values in general, higherorder terms can be neglected during the error synthesis modeling.Therefore, it is more convenient to omit the higher order terms firstduring the error modeling of each drive axis as above, rather than toexclude the higher order terms after the final error synthesis model isproduced.

The multi-axis controlled machine is embodied by continuously connectinga number of drive axes (rotary axes and linear axes) to each other. Whenviewed in the connecting direction, the axes may be connected in forwarddirection with respect to the reference coordinate system or in backwarddirection with respect to the home position of the machine.

N_(F) numbers of drive axes of the multi-axis controlled machine,continuously connected from the reference coordinate system {0}, areexpressed by models of matrix multiplication as in Formula 9, and theresult of this modeling is defined by Formula 10 below.

Where τ_(i) ^(j): transform matrix from {j} coordinate system to {i}coordinate system

N_(F): number of drive axes connected in forward direction in referencecoordinate system

o_(i: 3×1) vector representing offset between drive axes

S_(i): 3×3 skew matrix representing squareness between drive axes

t_(i): 3×1 vector representing translation input value of linear slide

A_(i): 3×3 rotation conversion matrix representing rotation input valuesof rotary table

d_(i): 3×1 vector representing position error at point of input value ofdrive axis

E_(i): 3×3 skew matrix representing angular error at a point of inputvalue of drive axis

i, j, k are dummy variables as the indices of summation operator (Σ) orproduct operator (II).

where

${{\sum\limits_{i = m}^{n}\; x_{i}} = {x_{m} + x_{m + 1} + \ldots + x_{n - 1} + {x_{n}\mspace{11mu}\left( {{{or}\mspace{14mu}{\prod\limits_{i = m}^{n}\; x_{i}}} = {x_{m}x_{{m + 1}\;}\ldots\mspace{14mu} x_{n - 1}x_{n}}} \right)}}}\mspace{11mu}$represents summation (or product) operator of adding (or multiplying) asequence of terms as a mathematical notion, x means an indexed variablerepresenting each successive term in the series, i represents the indexof summation as a dummy variable, and m and n indicate the lower andupper bounds of summation.

FIG. 10 b shows local coordinate systems about N_(F) number of driveaxes, which are connected continuously in backward direction withrespect to the reference coordinate system. For the kinematic errormodeling of the drive axes, which are connected in backward direction,it is essential to produce an inverse matrix of each parameter. Inpractice, inverse conversion of a matrix, which includes unknownvariables, using a computer is especially difficult and consumes amassive amount of time. Therefore, technology capable of simplifying theinversion conversion is necessary. First, inversion matrixes of TM, OM,and DM, which include position vector components, have negative positionvectors as expressed in Formula 11 below.(OM)⁻¹ ≡[I¦o] ⁻¹ =[I¦−o](TM)⁻¹ ≡[I¦t] ⁻¹ =[I¦−t](DM)⁻¹ ≡[I¦d] ⁻¹ =[I¦−d]  Formula 11

Since AM is an orthogonal matrix, the inversion matrix of AM, whichrepresents the rotation of the rotary drive axis, can be produced by atranspose matrix as in Formula 12 below.∴(Am)⁻¹ ≡[A¦0]⁻¹ =[A ^(T)¦0]  Formula 12

Finally, since the rotation components of EM and SM are decomposed by asum of skew matrices as in Formulae 2 and 5 above, the inverse matricesof EM and SM are produced by Neumann series as in Formula 13 below.(EM)⁻¹ ≡[I+E¦0]⁻¹ ≈[I−E¦0](SM)⁻¹ ≡[I+S¦0]⁻¹ =[I−S¦0]  Formula 13

It is possible to perform kinematic error modeling on the drive axes,which are connected in both directions, by applying the inverse matricesin Formulae 11 and 12. This can also be rearranged by performingmathematical modeling again, and be derived into Formula 13 below. Inthis deriving process, it is possible to improve the efficiency byomitting higher order terms of errors before producing a final model.

Where τ_(i) ^(j): transform matrix from {j} coordinate system to {i}coordinate system

N_(I): number of drive axes connected in backward direction in referencecoordinate system

o_(i): 3×1 vector representing offset between drive axes

S_(i): 3×3 skew matrix representing squareness between drive axes

t_(i): 3×1 vector representing translation input value of linear slide

A_(i): 3×3 rotation conversion matrix representing rotation input valuesof i th rotary table

d_(i): 3×1 vector representing position error at point of input value ofdrive axis

E_(i): 3×3 skew matrix representing angular error at a point of inputvalue of drive axis

T: Transpose of a matrix

i, j, k are dummy variables as the indices of summation operator orproduct operator.

The multi-axis controlled machine can be connected in forward directionor backward direction with respect to the reference coordinate system.Thus, for the purpose of versatility, the present invention defines anerror synthesis model generalized to multi-axis controlled machines byperforming mathematical modeling in consideration of NF number of driveaxes, which are connected in forward direction with respect to thereference coordinate system, and NI number of drive axes, which areconnected in backward direction with respect to the reference coordinatesystem. The generalized error synthesis model is expressed finally byFormula 15 below. This deriving process can also improve efficiency byomitting higher order terms of errors before producing the final model.

where τ_(i) ^(j): transform matrix from {j} coordinate system to {i}coordinate system,

N_(F): number of drive axes connected in forward direction in referencecoordinate system

N_(I): number of drive axes connected in backward direction in referencecoordinate system

o_(i): 3×1 vector representing offset between drive axes

S_(i): 3×3 skew matrix representing squareness between drive axes

t_(i): 3×1 vector representing translation input value of linear slide

A_(i): 3×3 rotation conversion matrix representing rotation input valuesof rotary table

d_(i): 3×1 vector representing position error at point of input value ofdrive axis

E_(i): 3×3 skew matrix representing angular error at a point of inputvalue of drive axis

T: Transpose of a matrix

i, j, k are dummy variables as the indices of summation operator orproduct operator.

Here, τ_(N) _(I) ^(N) ^(F) is an HTM that represents the posture of thelocal coordinate system that is located at the terminal end of theforward direction, when viewed from the local coordinate system that islocated at the terminal end of the backward direction. Rotationindicates a 3×3 sub-matrix that represents rotation components in theposture between the two coordinate systems, and Translation indicates a3×1 position vector that represents translation components between thetwo coordinate systems. In addition, N_(I) indicates the number of thedrive axes in backward direction, and N_(F) indicates the numbers of thedrive axes in forward direction.

For example, a multi-axis machine tool has an N_(I) number of drive axesfrom the reference coordinate system to a workpiece coordinate systemand an N_(F) number of drive axes from the reference coordinate systemto tool coordinate system. In addition, Rotation and Translationindicate the orientation and position of the tool coordinate systemviewed from the workpiece coordinate system.

The error synthesis model generalized as above can be used in anystructure of combination, which includes linear and rotary axes, byapplying geometric models to any axes. In addition, it is possible togeneralize this model by simultaneously considering forward and inversekinematic chains. Furthermore, the modeling can be performed by omittinghigher order terms of errors in advance to avoid tedious calculations,and the final error synthesis model can be produced in both forward andbackward directions irrespective of the number of axes.

For example, when an error synthesis model of the five-axis machine toolshown in FIG. 5 is produced, kinematic error models are defined as inFormula 16 below. One of the kinematic error models includes error ofthe multi-axis controlled machine, whereas the other kinematic errormodel does not include error of the multi-axis controlled model. InFormula 16 below, Nominal indicates a synthesis model that represents anideal state without error, whereas Actual indicates an actual errorsynthesis model with error.τ_(p) ^(t)|_(Nominal)=(τ_(R) ^(Y)τ_(Y) ^(A)τ_(A) ^(C)τ_(C) ^(p))⁻¹τ_(R)^(X)τ_(X) ^(Z)τ_(Z) ^(t)|_(Nominal)τ_(p) ^(t)|_(Actual)=(τ_(R) ^(Y)τ_(Y) ^(A)τ_(A) ^(C)τ_(C) ^(p))⁻¹τ_(R)^(X)τ_(X) ^(Z)τ_(Z) ^(t)|_(Actual)  Formula 16

In addition, when the final posture error of the five-axis machine toolis produced using the above-described generalized error synthesis model,it is produced as in Formula 17 below.

The present invention intends to estimate the error of multi-axiscontrolled machines using the error synthesis model generalized asabove.

FIG. 11 is a block diagram showing the configuration of the apparatusfor estimating error in a multi-axis controlled machine according to anexemplary embodiment of the invention. Referring to FIG. 11, theapparatus for estimating error in a multi-axis controlled machineaccording to an exemplary embodiment of the invention includes a machineconfiguration definition module 10, an error synthesis module 20, and anerror mapping module 30.

The machine configuration definition module 10 defines a topology ofmachine, that is, the environment of the multi-axis controlled machine,which is subjected to error estimation. More specifically, the machineconfiguration definition module 10 serves to define the drive axes ofthe multi-stage controlled machine, the posture of tools, the setting ofsquareness error, offset distances between axes, and the like.

The error synthesis module 20 defines geometric errors appropriate tothe machine configuration of the multi-axis controlled machine andperforms modeling on the parameters of error measurement data. In thisfashion, the error synthesis module 20 generates a final error synthesismodel by applying the generalized error synthesis model, as expressed inFormula 14, to the structure of the multi-axis controlled machine, whichis subjected to error estimation.

The error mapping module 30 creates a virtual coordinate, which includeserror of a designated shape or position, using the error synthesis modeldefined by the error synthesis module 20 and generates an error map,which visually expresses the error.

Accordingly, the user can predict and estimate the final error, whichoccurs from the multi-axis controlled machine, using the error mapgenerated by the error mapping module 30 and compensate for the errorusing the result of the error estimation.

The detailed operations and functions of the individual modules can beunderstood more fully with reference to the error estimation method,which is described hereinafter.

FIG. 12 is a flowchart showing the method for estimating error in amulti-axis controlled machine according to an exemplary embodiment ofthe invention.

Referring to FIG. 12, the method for estimating error in a multi-axiscontrolled machine according to an exemplary embodiment of the inventionincludes processes of machine configuration definition S301, errorsynthesis modeling S302, production of imaginary points including errorS303, and error mapping S304.

The machine configuration definition process S301 is a process that isexecuted by the machine configuration definition module 10 shown in 11.The machine configuration definition process S301 defines the structureof the multi-axis controlled machine in order to generate an errorsynthesis model of the same machine. Specifically, the machineconfiguration definition process S301 defines drive axes of themulti-axis controlled machine, the posture of a tool, offset distancesbetween the drive axes, the setting of squareness error, and the like.The definition of the drive axes of the multi-axis controlled machine isset by sequentially arranging the axes, which are formed between thereference coordinate system {R} and the workpiece coordinate system {P}and between the reference coordinate system {R} and the tool coordinatesystem {T}, and can be expressed by the form of {p, . . . , r, . . . ,t}. Here, P indicates a workpiece, r indicates the reference coordinatesystem, and t indicates the tool. The drive axes formed between theworkpiece and the tool can be defined by sequentially inserting thedrive axes, which are formed between the reference coordinate system andthe workpiece, in between p and r and the drive axes, which are formedbetween the reference coordinate system and the tool, in between r andt. For example, in the case of the five-axis machine tool shown in FIG.5, the axes are constructed in the order of: “reference coordinatesystem→Y axis→A axis→C axis→workpiece” and “reference coordinatesystem→X axis→Z axis→tool.” Thus, the definition of the drive axes is{p, c, a, y, r, x, z, t}. In addition, the definition of the posture ofthe tool represents the orientation consistent with the posture of thetool 2 and is expressed by one selected from among x, y, and zorientations. In the case of the five-axis machine tool shown in FIG. 5,the posture of the tool is expressed by one selected from among x, y,and z orientations. In addition, the definition of the offset distancesbetween the drive axes is to set the distance values between the driveaxes, which are set at the time of designing or manufacturing, and theoffset errors, which are obtained by measurement, by dividing thedistance values and the offset errors in x, y, and z orientations. Thisprocess can be facilitated by user input. In this process, tool offsetcan also be considered in the input.

In addition, the machine configuration definition process S301 serves todefine a reference axis and a sub-reference axis for squareness error.In the case of the machine tool, this process is executed as describedabove with respect to the squareness parameter when corresponding axesof the three linear axes y, y, and z are input respectively.

The machine configuration definition process S301 as described above canbe executed when the user sequentially inputs the drive axes, toolposture, and offset distance through input windows, which occursequentially.

After the structure of the multi-axis controlled machine is defined, theprocess of error synthesis modeling S302 is performed on the multi-axiscontrolled machine by applying the above-defined mechanistic structureand the error measurement data to the generalized error synthesis model,which is calculated by the above-described technology.

The process of error synthesis modeling S302 is executed by the errorsynthesis module 20. This process applies the generalized errorsynthesis model, as in Formula 14 above, to the multi-axis controlledmachine by defining the parameters according to the multi-axis shape,which is based on the configuration of the multi-axis controlled machineas defined above, and performing parametric modeling on the errormeasurement data, which is measured by an error measurement device orthe like.

FIG. 13 is a flow diagram showing error synthesis modeling in detailaccording to an exemplary embodiment of the invention.

With reference to FIG. 13, the process of error synthesis modeling 302is described in more detail. The process of error synthesis modeling 302includes steps S401 and S402 in order to generate the error synthesismodel. In S401, local coordinate systems of respective links of thecorresponding multi-axis controlled machine are defined with referenceto the mechanistic in the foregoing process S301, and in S402,parameters corresponding to the values of drive axes, the relationshipsof the drive axes, and geometric errors are defined based on the localcoordinate systems. The errors defined herein are the same as in Table 1above.

Afterwards, the process of error synthesis modeling 302 calculates, inS403, sub-matrices and vectors by applying the above-defined parametersto the generalized error synthesis model as in Formula 14 above in orderto generate the final error synthesis model. In the five-axis machinetool, the result of this calculation is, for example, the same as theresult in Formula 15 above.

Afterwards, in S404, the measurement data of the errors are inputsequentially through the input windows. In this case, individualdistance errors and error measurement values are measured as errorvalues of target distances and angles and are managed according to errortypes, error orientations, and drive axes. For example, the individualdistance errors and the individual angular errors are measured accordingto the drive axes and are divided in x, y, and z orientations when theyare presented. Here, a specific type of error in one orientation of onedrive axis is composed in one file (e.g., an Excel file) as shown inFIG. 15 a.

In addition, in S404, parametric modeling is performed on the errormeasurement data. This process is enabled by performing curve fitting onmeasurement points, in which the user can select an object curve modelfrom polynomial function, trigonometric function, or the like.

In addition, in S405, the final error synthesis model of the multi-axiscontrolled machine having the above-defined structure is generated byapplying all of the errors, on which the parametric modeling wasperformed, to the generalized error synthesis model, which wascalculated as above.

After the final error synthesis model is generated according to theshape of the multi-axis controlled machine through these processes, theprocess of producing imaginary points including error S303 is performed.

In the process of producing imaginary points including error S303, whenany shape to be processed (i.e., a processing shape) for errorestimation of the multi-axis controlled machine is set and ideal pointson a route of any processing shape are generated, imaginary pointsincluding errors are produced with respect to individual routes usingthe generalized error synthesis model. Afterwards, the process of errormapping S304 generates an error map, by which the error can beestimated, by connecting the above-produced imaginary points. In theerror mapping, an ideal shape, which does not include the error, is alsopresented so that the state of the geometric error and the performanceof the machine can be easily estimated.

FIG. 14 is a flow diagram showing the error mapping in detail accordingto an exemplary embodiment of the invention.

Referring to FIG. 14, in the process of producing imaginary pointsincluding error S303, a CAD shape to be estimated is designated ordirectly input by the user.

Next, tool path coordinates of individual drive axes have to begenerated according to the above-described shape. In the case of amachine tool, these processes can be obtained using CAM software, andinverse kinematics calculation is applied in the case of a five-axismachine tool. According to an exemplary embodiment of the invention,this process basically provides the step of processing a variety ofshapes, such as a hemisphere or a frustum of a cone according to theinternational standard SNA979, in S501 and the step of generating toolpath of translation in S502. In S503, this process also producescoordinates of processing tool path for other shapes by inputting motionparameters.

For example, as shown in FIG. 1, in the case of the five-axis machinetool having x, y, z, a, and c axes, coordinate values for the five axesare input. Here, units are given as options for individual drive axesand the user is allowed to select a unit that he/she intends. In thecase where error information, which occurs in the processing of acertain shape, is intended, listed coordinates presenting a designatedprocessing shape are input. The input can be formed as selecting a filein which listed coordinates of the machine are input (e.g., an Excelfile). The file is composed with reference to a data structure. If thelisted coordinates of the machine are input, the coordinates ofindividual axes are diagramed according to the numbers of the coordinateso that the user can inspect the continuity of the drive axes or thelike.

As described above, the motion parameters of the multi-axis controlledmachine are defined in S503. Next, this process produces, in S504,imaginary points including error of individual points in the path byapplying the input coordinates of the machine to the error synthesismodel defined for the multi-axis controlled machine, the error synthesismodel produced in the process of error synthesis modeling S302.

This process generates, in S505, an error map for the purpose ofcomparison by shaping imaginary points including the produced errors.Here, since an error vector has a very small value when compared to ashape coordinate, the user is allowed to select an error rate tovisualize the error map so that the error can be weighted or not. Forexample, if the error rate is 1, the error is not weighted. IF the errorrate is greater than 1, the error is weighted in proportion to the errorrate.

Meanwhile, if error information for one coordinate of the machine isrequested rather than that for a route including a plurality ofpositions, the processing shape is not designated but one coordinate forthe machine is allowed to be input. In this case, the error map isexpressed as shown in FIG. 16 so that ideal points of the machine arecompared with imaginary points including error. In this error map, thedistances between the ideal points of the machine and the imaginarypoints including error can be obtained.

When the listed coordinates of the machine are selected, the error mapis expressed as in FIG. 17( a), thereby making it possible to enablevisual error estimation across the entire coordinates. In addition, asshown in FIG. 17( b), error values are listed according to the numbersof coordinates so that the degrees of the errors according to thecoordinates can be produced.

In addition, it is possible to designate a certain processing shape andto estimate error occurring when a workpiece is processed in this shape.This is shown, by way of example, in FIGS. 18 and 19.

FIG. 18 is a diagram showing an example of an error map, which showserror distribution occurring when a workpiece is processed in the formof a frustum of a cone according to NAS 979. As shown in the figure,errors are shown according to the coordinates of the frustum of a cone.

FIG. 19 is a diagram showing an error map when a workpiece is processedin the form of a hemisphere. In this figure, errors are shown accordingto the coordinates of the processing tool path of the hemisphericalshape.

The foregoing descriptions of specific exemplary embodiments of thepresent invention have been presented for the purposes of illustrationand description. They are not intended to be exhaustive or to limit theinvention to the precise forms disclosed, and obviously manymodifications and variations are possible in light of the aboveteachings. The exemplary embodiments were chosen and described in orderto explain certain principles of the invention and their practicalapplication, to thereby enable others skilled in the art to make andutilize various exemplary embodiments of the present invention, as wellas various alternatives and modifications thereof. It is intended thatthe scope of the invention be defined by the Claims appended hereto andtheir equivalents.

What is claimed is:
 1. A method for estimating error in a multi-axiscontrolled machine comprising: defining the structure of the multi-axiscontrolled machine subjected to error estimation; determining localcoordinate systems for each link of the multi-axis controlled machine;defining parameters, which represent behaviors of drive axes havinggeometric error and relationships between the drive axes according tothe defined structure of the multi-axis controlled machine; adding thedefined parameters by applying the parameters to a generalized errorsynthesis model; performing parametric modeling on error measurementdata, which are measured from the multi-axis controlled machine;generating an error synthesis model of the multi-axis controlled machineby applying the result of parametric modeling in response to the resultof the adding; wherein the generalized error synthesis model isconsisting of: kinematic modeling of drive axes, which include kinematicerror models of linear and rotary axes; modeling one or more drive axesby arranging the drive axes in forward or backward direction withrespect to a reference coordinate system; and omitting higher orderterms of errors from the kinematic error modeling, which includesarranging the drive axes; and wherein the generalized error synthesismodel is expressed by formulas:

where τ_(i) ^(j) is a transform matrix from {j} coordinate system tocoordinate system, N_(F): number of drive axes connected in forwarddirection in reference coordinate system N_(I): number of drive axesconnected in backward direction in reference coordinate system o_(i):3×1 vector representing offset between drive axes S_(i): 3×3 skew matrixrepresenting squareness between drive axes t_(i): 3×1 vectorrepresenting translation input value of linear slide A_(i): 3×3 rotationconversion matrix representing rotation input values of rotary tabled_(i): 3×1 vector representing position error at point of input value ofdrive axis E_(i): 3×3 skew matrix representing angular error at a pointof input value of drive axis T: Transpose of a matrix${\sum\limits_{i = m}^{n}\; x_{i}} = {x_{m} + x_{m + 1} + \ldots + x_{n - 1} + {x_{n}:}}$summation operator of adding a sequence of terms${\prod\limits_{i = m}^{n}\; x_{i}} = {x_{m}x_{{m + 1}\;}\ldots\mspace{14mu} x_{n - 1}{x_{n}:}}$product operator of multiplying a sequence of terms i, j, k: dummyvariables as the indices of summation operator or product operator. 2.The method according to claim 1, wherein defining the structure of themulti-axis controlled machine defines selecting one or more axis fromthe group consisting of the drive axes of the multi-axis controlledmachine, a posture of a tool, offset distances between the drive axes,and a setting of squareness error.
 3. The method according to claim 2,wherein defining the drive axes of the multi-axis controlled machinecomprises sequentially arranging the drive axes, which are oriented inbackward and forward directions in a reference coordinate system of themulti-axis controlled machine.
 4. The method according to claim 1,further comprising: creating a virtual coordinate, which includes errorof a designated shape to be processed or at a position to be confirmed,using the error synthesis model; and generating an error map, whichvisually expresses the created imaginary point.
 5. The method accordingto claim 4, wherein creating a virtual coordinate comprises: determiningthe designated shape to be processed or the position to be confirmed;generating a processing route of the multi-axis controlled machine forthe designated shape to be processed or the position to be confirmed;generating or inputting a motion parameter about each of the drive axesof the multi-axis controlled machine along the route processing of themulti-axis controlled machine; and creating a virtual coordinate havingerror for each of points of the processing route using the errorsynthesis model.
 6. The method according to claim 4, wherein thedesignated shape to be processed is a frustum of a cone or a hemisphere.7. An apparatus for estimating error in a multi-axis controlled machine,comprising: a machine configuration definition module for defining atopology of a machine subjected to error estimation; an error synthesismodule for defining parameters, which represent behaviors of drive axeshaving geometric error and relationships between the drive axesaccording to the defined structure of the multi-axis controlled machine,adding the defined parameters by applying the parameters to ageneralized error synthesis model, and generating an error synthesismodel of the multi-axis controlled machine by applying a result ofparametric modeling in response to a result of the adding; and an errormapping module for creating a virtual coordinate, which includes errorof a designated shape to be processed or at a position to be confirmed,using the error synthesis model, and generating an error map, whichvisually expresses the produced imaginary point; wherein the generalizederror synthesis model is derived by selecting one or more from the groupconsisting of: kinematic modeling of drive axes, which include kinematicerror models of linear and rotary axes; modeling one or more drive axesby arranging the drive axes in forward or backward direction withrespect to a reference coordinate system; and omitting higher orderterms of errors from the kinematic error modeling, which includesarranging the drive axes; and wherein the generalized error synthesismodel is expressed by formulas:

where τ_(i) ^(j) is a transform matrix from {j} coordinate system to {i}coordinate system, N_(F): number of drive axes connected in forwarddirection in reference coordinate system N_(I): number of drive axesconnected in backward direction in reference coordinate system o_(i):3×1 vector representing offset between drive axes S_(i): 3×3 skew matrixrepresenting squareness between drive axes t_(i): 3×1 vectorrepresenting translation input value of linear slide A_(i): 3×3 rotationconversion matrix representing rotation input values of rotary tabled_(i): 3×1 vector representing position error at point of input value ofdrive axis E_(i): 3×3 skew matrix representing angular error at a pointof input value of drive axis T: Transpose of a matrix${\sum\limits_{i = m}^{n}\; x_{i}} = {x_{m} + x_{m + 1} + \ldots + x_{n - 1} + {x_{n}:}}$summation operator of adding a sequence of terms${\prod\limits_{i = m}^{n}\; x_{i}} = {x_{m}x_{{m + 1}\mspace{11mu}}\ldots\mspace{14mu} x_{n - 1}{x_{n}:}}$product operator of multiplying a sequence of terms i, j, k: dummyvariables as the indices of summation operator or product operator. 8.The apparatus according to claim 7, wherein the mechanistic structuredefinition module defines one or more axis selected from the groupconsisting of the drive axes of the multi-axis controlled machine, aposture of a tool, a setting of squareness, and offset distances betweenthe drive axes error.
 9. The apparatus according to claim 8, wherein themachine configuration definition module defines the drive axes of themulti-axis controlled machine by sequentially arranging the drive axes,which are oriented in backward and forward directions in a referencecoordinate system of the multi-axis controlled machine.
 10. Theapparatus according to claim 7, wherein the designated shape to beprocessed is a frustum of a cone or a hemisphere.
 11. The apparatus ofclaim 7, wherein the generalized error synthesis model is derived by:kinematic modeling of drive axes, which include kinematic error modelsof linear and rotary axes; modeling one or more drive axes by arrangingthe drive axes in forward or backward direction with respect to areference coordinate system; and omitting higher order terms of errorsfrom the kinematic error modeling, which includes arranging the driveaxes.